Alternating-direction implicit method

In mathematics, the alternating direction implicit (ADI) method is a finite difference method for solving differential equations. It is most notably used to solve the problem of heat conduction or solving the diffusion equation in two or more dimensions.

The traditional method for solving the heat conduction equation is the method of Crank-Nicolson. This method is implicit, but has an unaffordable stability criterion in two or more dimensions.

The method

Consider the linear diffusion equation in two dimensions,

${\displaystyle {\partial u \over \partial t}=\left({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}\right)=(u_{xx}+u_{yy})\quad }$ 

The implicit Crank-Nicolson method produces the following finite difference equation:

${\displaystyle {u_{ij}^{n+1}-u_{ij}^{n} \over \Delta t}={1 \over 2}\left(\delta _{x}^{2}+\delta _{y}^{2}\right)\left(u_{ij}^{n+1}+u_{ij}^{n}\right)}$ 

where ${\displaystyle \delta _{p}^{2}}$  is the central difference operator for the p-coordinate After performing a stability analysis, it can be shown that this method will be stable as long as

${\displaystyle {\Delta t \over (\Delta x)^{2}}+{\Delta t \over (\Delta y)^{2}}<{1 \over 2}.}$ 

This an unaffordable numerical stability criterion.

The idea behind the ADI method is to split the finite difference equations into two, one with the x-derivative taken implicitly and the next with the y-derivative taken implicitly,

${\displaystyle {u_{ij}^{n+1/2}-u_{ij}^{n} \over \Delta t/2}=\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n}\right)}$ 

${\displaystyle {u_{ij}^{n+1}-u_{ij}^{n+1/2} \over \Delta t/2}=\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n+1}\right)}$ 

It can be shown that this method is unconditionally stable. There are more refined ADI methods such as the methods of Douglas^[1], or the f-factor method^[2] which can be used for three or more dimensions.

1. Douglas, J. "Alternating direction methods for three space variables," Numerische Mathematik, Vol 4., pp 41-63 (1962)
2. Chang, M.J. et al. "Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems", Numerical Heat Transfer, Vol 19, pp 69-84, (1991)